Final answer:
To evaluate the double integral ∫∫ℝ(5x+6y)²dA, we set the iterated integral's limits as A = 0, B = -1, C = 1 + x, and D = 1. We integrate first with respect to y, then with respect to x. The integration involves expanding the square of the function and integrating term by term.
Step-by-step explanation:
The question involves setting up and evaluating a double integral over a triangular region R with vertices at (-1,0), (0,1), and (1,0). The integrand is a function of x and y, specifically (5x+6y)². To solve this problem, it is necessary to determine the limits of integration for both the x and y integrals, taking into account the geometry of R.
First, we notice that the triangle is symmetrical about the y-axis, so the x-limits of integration will range from -1 to 1. The y-limits are a bit more involved. For a given x-value, y starts at 0 (at the base of the triangle) and goes up to the line connecting the points (-1, 0) and (0, 1), which has the equation y = 1 + x. So, our y-limits will be from 0 to 1 + x.
B and D are the limits for the outer integral (x-integral), so B = -1 and D = 1. A and C are the limits for the inner integral (y-integral), so A = 0 and C = 1 + x. Therefore, the iterated integral is:
∫-1∫1 ∫0∫1+x (5x+6y)² dy dx
To provide the correct answer by solving, you would integrate first with respect to y, from A to C, and then with respect to x, from B to D. The integration itself can be fairly complex, involving expanding the square and integrating term by term.